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\section{Question wording of the Census Race Question}\label{sec:sup_census_race}

Our measure of objective context used the 2006 Canadian Census reports of responses to the
following two questions:

\begin{quote}
(1) Mark more than one or specify, if applicable. This information is collected
in accordance with the Employment Equity Act and its Regulations and Guidelines
to support programs that promote equal opportunity for everyone to share in the
social, cultural, and economic life of Canada.  (1) Is this person an Aboriginal
person, that is, First Nations (North American Indian), Metis or Inuk (Inuit)?

(2) Is this person: White, South Asian (e.g., East Indian, Pakistani, Sri
Lankan, etc.), Chinese, Black, Filipino, Latin American, Arab, Southeast Asian
(e.g., Vietnamese, Cambodian, Malaysian, Laotian, etc.), West Asian (e.g.,
Iranian, Afghan, etc.), Korean, Japanese, Other (specify)?
\end{quote}

\section{Measures of Subjective Context}

We considered multiple ways to measure people's perceptions of the numbers of
visible minorities around them. Version 1 simply adds up all responses for the
VM groups. Version 2 runs from 0 to 1, where 0 represents 0 percent and 1
represents all responses that added up to 100 percent or more. Version 3
recalculates responses such that a respondents' answers for all groups totaled
100 percent; in other words, a respondent who said her community was 50 percent
Latin American, 50 percent Black, and 50 percent White would have a VM score of
67 percent. We chose to use Version 1 in the analyses presented. Version 2
makes the display of information easier and down-weights outliers, but it
throws away information distinguishing respondents whose estimates total 110
percent and 500 percent, for example. Version 3 helps address the issue of
innumeracy --- and the problem that many ordinary citizens do not realize that
percentages should total 100 --- but it makes the assumption that the
ethnic/racial groups listed are mutually exclusive and that they are the only
ones that count.

\section{Design Details}\label{sec:app_design_details}

The MLCC Study involved roughly 7800 English speaking Canadians spread across
all the provinces of Canada. We worked to isolate comparisons based on
perceptions of the percent visible minority (VM) in local areas from Census
measurements of percent VM in local areas in two main ways: (1) by matching
people into pairs based on the percent VM in the Canadian Census dissemination
area (DA) measured in 2006, the change in percent VM in the DA between 2006 and
2016, and census defined urbanicity of the municipality (the CSD) and (2) by
restricting comparisons to people who lived in the same DA.\footnote{This
document can be rebuilt from the unix command line using R and LaTeX and the
GNU Make system by downloading the reproduction archive (insert link upon
publication). at  \url{http://github.com/bowers-illinois-edu/wong_bjps_2025}. }

We originally had intended to use both 2011 and 2006 census data to look at
contemporaneous and changes in diversity. However, in 2011, the long-form of
the Census --- which is where Canadians are asked about their ethnicity and
race --- became voluntary in the newly renamed National Household Survey
\citep{thompson2010politics}. The response rate dropped 25 percentage points.
The Census summaries for small geographic units (such as dissemination areas)
were made particularly imprecise and/or are missing given this change in the
Census \citep{Sheikh2013, grant2015damage}. Since the survey occurred in 2012,
we hope that by matching on change between 2006 and 2016, we can capture some
of the difference in visible minority population between that measured in 2006
and what would have been measured in 2011 if the visible minority items had
been used universally.

\subsection{Matching on Visible Minority in the DA, Perceptions of Own Local
Community.} \label{sec:app_design_anyDA}

This design compared perceptions of context with subjective boundaries within
pairs matched on objective context. That is, the ``effect of perceptions'' in
this design was defined as a comparison of perceptions of the ethnic
composition of the local community maps drawn by the respondents. Before
matching in this design, we excluded the roughly 1425 respondents who did not
draw a map or report on their perception of visible minorities in their map.
The pair-creation process yielded 3772 respondents in 1886 pairs
representing 3098 DAs.

% see Analysis/supp_desc_new.Rout for the numbers
% > ## How many people did we exclude either for not providing any perceptions or
% > ## not drawing a map?
% > nrow(big_wrkdat_thin)
% [1] 7811
% > nrow(wdat0)
% [1] 6386
% > nrow(big_wrkdat_thin) - nrow(wdat0)
% [1] 1425
% > ## Total number of respondents used in the matching after excluding a few
% > nrow(wdat0[!is.na(wdat0$pair), ])
% [1] 3772
% > ## Number of pairs
% > num_pairs
% [1] 1886
% > ## Number of DAs
% > num_das
% [1] 3098

\begin{figure}[H]
  \centering
    \includegraphics[width=\linewidth]{coefplot_anyDA_new.pdf}

    \caption{Average differences in outcome between the person perceiving more
      and the person perceiving fewer visible minorities in their own
      hand-drawn maps within pairs matched on \% VM in DA, change in \% VM
      2006--2016 in the DA, and urban-vs-rural classification of the CSD.
    Estimates (black dots) from multilevel models with crossed random effects
  for dissemination area and matched pair. Approximate confidence intervals
(horizontal lines) from a profiled likelihood approach \citep{bates2015lme4}.}

  \label{fig:app_coefplot_anyDA}
\end{figure}

%\clearpage
\subsection{Matching on Visible Minority in the DA, Perceptions of Own DA.}  \label{sec:app_design_DA}

This design assessed the influence of perceptions by comparing reported percent
visible minorities by respondents who were shown their Census DA as a polygon
overlaid on a Google map. Since we randomly assigned roughly 1/6 of the sample
to be exposed to and report on their DA (and other sixths to see and report on
other census geographies), and because we excluded respondents who did not
answer the perceptions question, this design used roughly 448
respondents out of a possible 650, with 443 DAs represented.

The matches again are homogeneous in objective context:  20 percent of the
matches differ by less than .001 on their DA level diversity; the median difference in
percentage visible minority between matches is 0.01 percent; 90 percent of
matches have a difference of less than 1 percentage points, and the maximum is
1.6 percentage points.  When it comes to population size of the DA, the median
difference is 182 people, 90\% of pairs differ by less than 1122 and the
maximum difference is 11,429.

% from Design/match_assess_DA_new.Rout
% 650 people who were assigned to see their DA and who had valid outcomes and perceptions
% 448 people matched
% 443 DAs
%> sapply(pairdiffs, function(x) {
%+   quantile(x, sort(unique(c(seq(0, 1, .1), seq(.9, 1, .01)))), na.rm = TRUE)
%+ })

%#       pair da_prop_vm_20pct_06 vm_change     vm da_pop_06 
%# 0%     1.0          0.00000000  0.000000 0.0100       0.0 
%# 10%   23.3          0.00000000  0.001581 0.0300      23.1 
%# 20%   45.6          0.00009896  0.005265 0.0660      46.0 
%# 30%   67.9          0.00044887  0.008501 0.1000      84.9 
%# 40%   90.2          0.00091541  0.011661 0.1300     128.4 
%# 50%  112.5          0.00143456  0.014479 0.1700     182.0 
%# 60%  134.8          0.00243473  0.017898 0.2100     249.2 
%# 70%  157.1          0.00357245  0.019897 0.2710     375.5 
%# 80%  179.4          0.00541113  0.022633 0.3400     586.4 
%# 90%  201.7          0.00901861  0.026221 0.4800    1122.8 
%# 91%  203.9          0.00908558  0.026390 0.4993    1324.0 
%# 92%  206.2          0.00941058  0.026802 0.5100    1454.0 
%# 93%  208.4          0.00991629  0.027148 0.5278    1565.4 
%# 94%  210.6          0.01078593  0.027474 0.5824    1804.7 
%# 95%  212.9          0.01112658  0.028284 0.6740    2072.7 
%# 96%  215.1          0.01123025  0.028435 0.7632    2452.2 
%# 97%  217.3          0.01163234  0.029130 0.9117    2638.6 
%# 98%  219.5          0.01268062  0.029422 1.0362    3827.6 
%# 99%  221.8          0.01409258  0.029688 1.3231    6011.4 
%# 100% 224.0          0.01626694  0.029860 4.1900   11429.0 


The analysis show us the same pattern of results as before:  those
who perceive a more diverse dissemination area within their pair tend also to
be those reporting diminished perceptions of social capital and collective
efficacy. 


\begin{figure}[H]
  \centering

    \includegraphics[width=\linewidth]{coefplot_DA_new.pdf}

  \caption{Average differences in outcome between the person perceiving more
    and the person perceiving fewer visible minorities in their own census DA
    within pairs matched on \% VM in DA, change in \% VM 2006--2016, and
    urban-vs-rural classification of the CSD from multilevel models with
    crossed random effects for dissemination area and matched pair.
  Approximate confidence intervals (horizontal lines) from a profiled
likelihood approach \citep{bates2015lme4}.}\label{fig:appObjDA}

\end{figure}

\subsection{Exact matching on DA, Perceptions of own Local Community}
\label{sec:app_design_sameDA}

% from Analysis/analysis_sameDA.R
% > ## Number respondents
% > nrow(sameDAdat)
% [1] 1894
% > ## Number of DAS
% > length(unique(sameDAdat$dauid))
% [1] 838
%
%> table(table(sameDAdat$dauid))
%%
%# 
%#   2   3   4   5   6   9 
%# 674 131  20   8   4   1 

This design assessed the effect of pseudoenvironments by comparing perceptions
of their own local communities by people who lived in the same DA. The MLCC
study had about 1894 respondents who shared a DA with at least one other
respondent and who had valid perceptions-of-their-own-maps data spread across
838 DAs. While there were 674 DAs containing only 2 respondents, there were 131
DAs containing 3 respondents, and 33 containing between 4 and 9 respondents.
Because we only compare people living in the same DA without further pairing,
we no longer have clustering by DA, so the results that we present below come
from linear regressions with fixed effects for DA.

Even when we match respondents who live in the same DA our overall story is
unchanged. Among individuals who live in exactly the same dissemination area,
those who see a more diverse community perceives lower social capital and
collective efficacy in their own hand-drawn local community than the people in
the same DA who perceive relatively less diversity in their local communities.

\begin{figure}[H]
  \centering

    \includegraphics[width=\linewidth]{coefplot_SameDA.pdf}

    \caption{Average differences in outcome between the person(s) perceiving
      more and the person(s) perceiving fewer visible minorities in their own
      hand-drawn map within groups (mostly pairs) living in the same
      dissemination area. Linear regression models with fixed effects for DA
      and HC2 robust standard errors.}\label{fig:appSameDA}

\end{figure}


%\clearpage
\subsection{Exact matching on DA, Perceptions of DA}
\label{sec:app_design_sameDAviewDA}

%From Analysis/sameDAviewDA.R
%> ## How many respondents?
%> nrow(sameDAviewDAdat)
%[1] 20
%> ## How many DAs
%> length(unique(sameDAviewDAdat$dauid))
%[1] 10
%> 
%

This design assessed the effect of subjective context by comparing perceptions
of the census DA between people living in the same DA. That is, in this design
we have people living in the same census location and evaluating the same
census object. The MLCC study included 20 people with these characteristics in
10 DAs who had valid outcome answers as well as valid answers about the
proportion visible minority of their Canadian Census DA polygon. Just by
chance, each of these DAs contained exactly 2 people. The plots below show the
mean differences in outcomes between the high and low perceiving person and the
intervals containing 75\% of these differences for the 10 DAs.

Even though there is a great deal of noise, the coefficients have the same sign
as in the previous analyses showing a negative relationship between perceptions
of diversity and social cohesion and collective efficacy.

\begin{figure}[H]
  \centering

    \centering
    \includegraphics[width=\linewidth]{coefplot_SameDAViewDA.pdf}

  \caption{Average differences in outcome between the person perceiving
    more and the person perceiving fewer visible minorities in the
    dissemination areas in which both people live. Intervals show the range of
    the central 75\% of the paired differences. Dots show the means of the
    paired differences.}\label{fig:appSameDAViewDA}

\end{figure}


%\clearpage
\subsection{Matching on Fractionalization Score of DA,
  Fractionalization Score of Perceptions of Own Local Community}
  \label{sec:app_design_anyDA_Diversity}

% from match_assess_anyDA_Diversity_new.Rout
%    # number of people in final design
% > nrow(wdat1)
% [1] 4330
% > # number of DAs
% > length(unique(wdat1$dauid))
% [1] 3599

This design assessed the effect of subjective context by comparing
fractionalization scores created from perceptions of people's own local
communities. We created the scores as the sum of the squared proportions of the
different racial and ethnic minorities measured at the DA level by the Census
using proportions of the following census categories: ``Black,""Chinese,""Latin
American,""South Asian," and ``Other Asian." This score is often know as the
Herfindahl-Hirschman index \cite{rhoades1993herfindahl}.

% from Data/get_and_setup_206_census_data.R
% census_data <- census_data %>% mutate(da_diversity_index = da_prop_other_asian^2 +
%   da_prop_chinese^2 +
%   da_prop_black^2 +
%   da_prop_latin^2 +
%   da_prop_south_asian^2)

The matches were created by pairing people living in DAs with very
similiar fractionalization scores within the Census urban-vs-rural
classification and requiring that pairs differ by less than 2.5 percentage
points in change in visible minority population between 2006 and 2016. These
restrictions left us with 4330 respondents matched into pairs representing 3599
DAs. The half of pairs in the final design were identical in the
fractionalization score and the maximum difference in this score (which runs
from 0 to .33) was .00009.


What is the effect of perceiving greater fractionalization on attitudes about
social capital? The results are similar to those we found
earlier:  the individual within each pair who sees greater heterogeneity
reports less social capital and collective efficacy.

\begin{figure}[H]
  \centering

    \includegraphics[width=\linewidth]{coefplot_anyDA_Diversity_new.pdf}

  \caption{Average differences in outcome between the person perceiving more
    and the person perceiving fewer visible minorities in the dissemination
    areas in which both people live. Estimates from multilevel models with
    crossed random effects for dissemination area and matched pair and
  covariance adjustment for proportion visible minority in the DA as of 2006.
Approximate confidence intervals (horizontal lines) from a profiled likelihood
approach \citep{bates2015lme4}.} \label{fig:appSameDAViewDiversity}

\end{figure}

\section{Analysis details: Crossed Random Effects versus Fixed Effects}\label{sec:app_mlm_fe_anyDA}

The paired designs presented in the main body of the text and in the preceding
sections of the appendix lead to analyses which summarize the relationship
between pair-wise differences in perceptions and pair-wise differences in
social cohesion and collective efficacy. We used multi-level modeling strategy
to further remove remaining relationships between objective and subjective
context within pairs and to calculate standard errors and confidence intervals
accounting for clustering by DA. Here we discuss our reasoning for choosing one
empirical strategy out of several and also compare the results between them.
The reader will notice that the substantive results and statistical
significance decisions remain the same across all of the approaches. We focus
our explorations here on the design with the most units, the design that we
discuss in the main body of the text in Figure 2. And we focus on the social
cohesion outcome. The point of this appendix is to illustrate that different
approaches to calculating standard errors and estimates do not change the
story.

% above Figure 2 refers to fig:coefplot_anyDA in the body.tex

Figure~\ref{fig:app_pairs_social_cohesion_anyDA} shows the raw relationship
between pair-differences in perceptions and pait-differences in the social
cohesion index. Across three different scatterplot smoothers (OLS, an outlier
robust linear smoother, and an outlier robust local non-linear smoother), we
see that as the difference in perceptions increases (such that one member of
the pair perceives a more heterogeneous community than the other) the
difference in social cohesion becomes more negative: the person perceiving a
more diverse community tends more and more to be the person responding with
lower values to the social cohesion scale. This figure suggests that a straight
line is a reasonable summary of relationship and that the slope of this line is
not driven by a few overly influential points.

\begin{figure}[H]
  \centering
    \includegraphics[width=.5\linewidth]{pairwise_plot_social_cohesion_anyDA_new.pdf}

    \caption{Difference in social cohesion between the higher and lower
      perceiver within pairs by difference in perceptions between the higher
      and lower perceiver. The relationship summarized by OLS (blue line), an
      outlier robust linear fit \autocite{koller2011sharpening}, and an outlier
      robust local smoother (loess, span=.5, degree=2). This plot excludes
    pairs where both people report the same perception.}

\label{fig:app_pairs_social_cohesion_anyDA}

\end{figure}

The analyses presented in the main body of the text and in the preceding
section improve on the simple smoother shown above in two ways.

First, in all of the designs other than the design with people living in
exactly the same DAs shown in  \ref{sec:app_design_sameDAviewDA}, members of
pairs are not exactly identical in terms of objective context even if the
matching process made them extremely similar. Since we are trying to isolate
the effects of perceptions from the effects of objective context, and following
the advice of \textcite{rubi:thom:comb:2000}, we adjust our estimates for any
remaining linear relationship between objective context and perceptions and
between objective context and the outcome. This is what
\textcite{rubi:thom:comb:2000} call ``covariance adjustment" and is often
referred to as ``controlling for" in political science. This strategy is
implemented below in Table~\ref{tab:app_mlm_vs_fe} in the column labeled ``FE"
for ``fixed effects" and the row labeled ``estimate" contains values calculated
using R code like the following.\footnote{Regressing pair-differences on
  pair-differences yields the same estimates as regressing individual level
  outcomes on individual level perceptions with fixed effects for pair. Since
  fixed effects models are common in political science, we label this model the
"fixed effects" model rather than the ``pair-differenced" model.}

% https://cran.r-project.org/web/packages/fixest/vignettes/standard_errors.html
\lstinputlisting[language=R]{estimation_example_fe.R}

Second, we improve the calculation of confidence intervals by using a
multi-level model. The fixed effects approach assumes that the observations are
independent across pairs when we know that they are not. The analogy in what we
are calling the fixed-effects  approach is to that of a randomized experiment
where first the units are placed into pairs, and then one person in each pair
is selected to have the higher of the two existing perception values. The
standard error in that design describes the variability we would expect in our
estimates from one hypothetical within-pair randomization to another one, where
each pair is independent of each other and randomization occurs only within
pairs.\footnote{See \textcite{gerber2012field}, Chapter 3 for an introduction
  to this perspective on design-based standard errors in block or
  pair-randomized experiments. The \lstinline[language=R]{lm_robust()} function
  by default calculates those design-based  standard errors
\autocite{Blair:2024aa}.}


% from Analysis/supp_desc_new.Rout
%> ## How many DAs shared across pairs?
%> num_das <- with(wdat0[!is.na(wdat0$pair), ], table(dauid))
%> summary(num_das)
%Number of cases in table: 3772 
%Number of factors: 1 
%> table(num_das)
%num_das
%   1    2    3    4    5    6    9 
%2533  496   47   12    5    4    1 
%> table(num_das > 1)
%
%FALSE  TRUE 
% 2533   565 
% 496+47 = 543 
%> table(num_das > 3)
%
%FALSE  TRUE 
% 3076    22 

In this design, although each person only appears in one pair, multiple pairs
can share the same value of objective context. In fact, although roughly 2533
respondents are the only survey respondents in their DA, about 543 share a
dissemination area with 1 or 2 other respondents, and 22 share their
dissemination area with more than 2 other survey respondents. Since pairs can
only contain 2 respondents, two people from the same dissemination area might
be paired with two other people from two different DAs or two people from the
same other DA. Since we have two kind of groups --- pairs and census
dissemination areas (DAs) --- and since those groups are not nested and the DA
groups vary in size, we worried that the fixed effects design-based standard
error from \lstinline[language=R]{lm_robust()} would be overly liberal because
it assumed an unrealistic independence across pairs.

Because this dependence is not a part of the design itself that we have
created, but arises from the fact that we want to adjust for objective context,
we depart from the standard practice of design-based statistical inference in
pair-matched randomized experiments \autocite{gerber2012field} and
non-randomized studies like ours \autocite{rosenbaum2020book} in this paper
although we show those results in the ``FE" columns in
Table~\ref{tab:app_mlm_vs_fe}.  Another tradition in the analysis of paired
experiments and observational studies uses probability models to describe the
outcome and relationships between predictors using a likelihood function such
that we imagine that the values of the social cohesion index (and/or community
efficacy index)  arise from some underlying stochastic process and also that
parameters of this model have their own generating processes that can reflect
dependencies across units. This approach, often known as the  model-based
approach, also allows us to make a model in which the responses to the social
cohesion index vary systematically by both pair and by the dissemination area.
This approach, known as a multi-level model or random-effects model, has been
proposed for use in matched studies \autocite{smith:1997} and as a part of a
Bayesian analysis of such designs in general \autocite{gelman2013bayesian}.

We present the multilevel model here in equation~\ref{eq:app_mlm_mod} for
respondents $i=1,\ldots,n$, matched pairs $s=1,\ldots,S$ and dissemination
areas $d=1,\ldots,D$. We write out the likelihood version below (but using some
Bayesian style notation to connect with the fully Bayesian
model).

\begin{eqnarray}
  y_{isd} \sim N(\mu_{isd},\sigma_{isd}) \nonumber \\
  \mu_{isd} = \alpha_{s}+\alpha_{d}+\beta_1 \text{perceptions}_{is} + \beta_2 \text{objective}_{id} \\ \label{eq:app_mlm_mod}
         \alpha_{s} \sim N(\gamma_{s,0},\sigma_{s}) \nonumber \\
         \alpha_{d} \sim N(\gamma_{d,0},\sigma_{d}) \nonumber
\end{eqnarray}

Equation~\ref{eq:app_bayes_mlm_mod} shows that the fully Bayesian version of
the model is nearly the same as the likelihood based model, but with different
prior distributions for the intercepts (using $t$-distributions and half-$t$
distributions ($t^{+}$) which are restricted to positive values rather than
Normal distributions to downweight extreme points), and with the addition of
prior distributions for other model parameters. This is the default prior
distribution used by the \lstinline[language=R]{brms} package.

\begin{eqnarray}
  y_{isd} \sim N(\mu_{isd},\sigma_{isd}) \nonumber \\
  \mu_{isd} = \alpha_{s}+\alpha_{d}+\beta_{1,isd} \text{perceptions}_{is} + \beta_{2,isd} \text{objective}_{id} \\ \label{eq:app_bayes_mlm_mod}
         \alpha_{s} \sim t(\gamma_{s,0},0,2.5) \nonumber \\
         \alpha_{d} \sim t(\gamma_{d,0},0,2.5) \nonumber \\
          \beta_{1,isd} \sim U(-\infty,\infty) \nonumber \\
          \beta_{2,isd} \sim U(-\infty,\infty)  \nonumber  \\
          \gamma_{s,0} \sim t^{+}(3,.5,2.5)  \nonumber  \\
          \gamma_{d,0} \sim t^{+}(3,.5,2.5)  \nonumber \\
          \sigma_{isd} \sim t^{+}(3,0,2.5) \nonumber
\end{eqnarray}

Table~\ref{tab:app_mlm_vs_fe} contains results from our use of
\lstinline[language=R]{brms}  \autocite{Burkner:2017aa} for the fully Bayesian
model and \lstinline[language=R]{lme4} \autocite{bates2015lme4} for the
likelihood based version with R using code like the following:

\lstinputlisting[language=R]{estimation_example_mlm.R}

Table~\ref{tab:app_mlm_vs_fe} shows the results used in
Figure~\ref{fig:app_coefplot_anyDA}, in the columns labeled ``MLM"  and
compares them to an alternative method that only accounts for the paired-design
using fixed effects (the columns labeled ``FE") as well as to the fully
Bayesian model, in the columns labeled ``Bayes MLM." The entries labeled
"standard error" and "95\% CI" show estimates of uncertainty for these
calculations. The standard errors and confidence intervals for the FE columns
are those that would be used in a large pair-randomized experiment
\autocite{gerber2012field} and are the defaults from the
\lstinline[language=R]{lm_robust()} command in R and are often known as the
"HC2" robust standard errors. They do not take into account dependence across
pairs within DA, but they do not rely on any assumptions about the data
generating process of the outcome (i.e., there is no Normality assumption in
regards the outcome that is required for these confidence intervals to be
valid). The multilevel model columns do involve assumptions about the data
generating processes (as shown in the preceding equations), however these
assumptions allow us to directly represent dependence across pairs by DA. The
confidence intervals for the MLM columns use the profiled-likelihood approach
recommended over the Wald-style intervals by \autocite{bates2015lme4},
\autocite{raudbryk02}, and \autocite{pinbates} for models with complex random
effects like ours. The confidence intervals for the Bayes MLM columns are the
boundaries of the 95\% highest posterior density regions for the posterior
distributions implied by the model and arising from MCMC sampling using the
\lstinline[language=R]{brms()} function to interface with the
\lstinline[language=R]{stan} Bayesian computation system. Standard errors for
the fully Bayesian model are the standard deviations of the posterior
distribution. Each approach has its own weaknesses and strengths, but the point
estimates in Table~\ref{tab:app_mlm_vs_fe}  shows that none differ in their
substantive interpretation.

\input{../Figures_Tables/appendix_xtab_anyDA_new.tex}

\paragraph{Intuition about likelihood profile confidence intervals}

A confidence interval for the effect of perceptions on social cohesion should
contain a range of values for this effect that are plausible and exclude those
values that are implausible. While many social scientists are used to
calculating a confidence interval as an unbiased estimate plus-or-minus roughly
2 times the standard error of that estimate, the intuition about plausible
values and the everyday use of confidence intervals can help us calculate those
intervals via another route: via hypothesis tests. First, recall that if the
95\% confidence interval contains 0, we know that the $p$-value for the test of
the hypothesis of no effects is larger than $p=.05$. That is, social scientists
commonly use confidence intervals as hypothesis tests. This use of confidence
intervals is perfectly correct because, in fact, a 95\% confidence interval is
a collection of hypotheses that would not be rejected at $\alpha=.05$ using a
given test statistic. This fact means that when, say, the standard
error of an estimator is difficult to calculate and/or the plausible range is
not necessarily plausibly symmetric around the estimate, one can generate
a confidence interval using a series of hypothesis tests.

The profile confidence interval arises from this logic and helps solve problems
that arise when the Hessian matrix for the log-likelihood function is
numerically unstable and/or difficult to calculate. These kinds of difficulties
arise from likelihood functions containing many parameters (such as likelihood
functions that arise from multilevel models with many random effects.)  The
idea is to use the Likelihood Ratio (LR) hypothesis test --- which tests the
hypothesis that a given change to a model improved its overall fit --- and to
search for the range of values for the parameter that do not significantly
improve the model fit over that of the maximum itself (which is the value of
the estimate reported as "Estimate" in the table.)


%\clearpage
\section{Discussion:  Alternative Explanations: What about personality?}%\label{sec:app_alt_personality}

One alternative explanation that we consider is that people's personalities are
an important factor in determining where they live, what they see in the maps
in their heads, and their attitudes about where they live. Unfortunately, we do
not have measures of personality in our survey. However, we hope in this case
that previous research on the Big 5 can help us assess this concern, at least
to a certain extent. Does personality affect residential choices?
\textcite{jokela2015geographically} show that openness to experience is most
strongly related to living in a city center where there is greater ethnic
diversity (and also higher population density, housing prices, and crime
rates).  Extraversion is also related to living in urban areas; in contrast,
living in more suburban neighborhoods is correlated with agreeableness and
conscientiousness.

Who has greater political knowledge (and by implication, will more accurately
perceive their contexts)?  \textcite{mondak2008framework} show that
the more knowledgeable people are individuals who are more open, less
conscientious, less extroverted, and possibly more neurotic.

When it comes to attitudes about immigration, 
\textcite{gallego2014big} find that agreeableness has the strongest relationship to
positive attitudes about immigrants' impact on a society. Openness to
experiences has a weak positive relationship to such attitudes, and this may
largely be driven by the fact that more educated individuals are more open and
also more tolerant about immigration.  Extraversion has no effect, and
neuroticism and conscientiousness are only weakly related to beliefs that there
are too many immigrants, that they do not help a neighborhood, and that they
should not be entitled to the same benefits as natives.

So, it is possible that in our matches, the respondent who scores higher on
openness to experience perceives fewer visible minorities in her local
community, (weakly) sees immigrants in a positive light, and also has lower
political trust \citep{mondak2008framework}; however, both types of respondents
are likely to live in similar types of areas, since openness is associated with
living in the city center.  The respondent in a match who is more conscientious
will perceive more minorities, and (weakly) perceive immigrants have a negative
impact, but again the pair likely live in similar areas, since the more
conscientious tend to live in the suburbs with fewer minorities. Finally,
agreeableness, which has the strongest predictive power of immigration
attitudes, has no relationship with political knowledge. 

More research is obviously needed, but at this point, we do not think there is
enough evidence to dispel our conclusion that pseudoenvironments matter for
attitudes about social capital above and beyond objective
environments.\footnote{It should be noted that the studies we cite here are not
  based on the same population that we use in our study although we think that
  the relationships driven by personality should not differ greatly across
  these studies: the residential choice study was based on a self-selected
  sample in the Greater London metropolitan area between 2009 and 2011; the
  knowledge study was based on surveys in select cities and counties in the US
  in 1998, 2004, and 2005; and the immigration attitudes study was based on a
representative national panel study of the Netherlands in 2007 and 2008.}

\subsection{Racial Resentment Question Wording}\label{sec:app_sym_rac}
Respondents were asked to agree or disagree with the following statements:

\begin{enumerate}
\item Irish, Italian, Jews and many other minorities overcame prejudice and
worked their way up. Other minorities should do the same without any special
favors.

\item It's really a matter of some people not trying hard enough; if racial and
ethnic minorities would only try harder they could be just as well off as
whites.

\item Government officials usually pay less attention to a request or complaint
from someone who is a racial or ethnic minority than from someone who is white.
\end{enumerate}

\section{Diagnostics about the paired design}\label{sec:app_design-diag}

Figure~\ref{nbmplot_anyDA} plots the respondents by their objective and
subjective context, connecting the individuals in each pair with a line
segment. Figure~\ref{nbmplot_anyDA} shows that respondents' views of who
lives around them --- particularly the proportion of ethnic outgroup members
--- vary within pairs of people living in nearly the same kinds of objective
contexts. The length of the lines from left to right shows that the
respondent who perceives greater diversity could be more or less accurate
than her match; similarly, both respondents could be quite inaccurate. The
fact that the lines are all relatively flat illustrates the similarity in
objective diversity within pairs (a sign that our design compares like with
like); the lengths of the lines tell us that perceptions of these matches
differ (a sign that perceptions are not a simple proxy for objective
context). The fact that most of the points are below the 45 degree line tells
us that the white respondents in the MLCC tended to report more visible
minorities in their hand-drawn local communities than were measured by the
Canadian Census in their DAs.


\begin{figure}[!ht]
 \centering
 \includegraphics[width=.75\linewidth]{nbmplot_anyDA_new.pdf}
 \caption{The Objective and Subjective Context of the Matched Pairs. The y-axis is the percentage of visible minorities in the Census DAs.
   The x-axis is the perception of the percentage of visible
   minorities in respondents' ``local communities.''}\label{nbmplot_anyDA}
\end{figure}

\clearpage

\section{Affirmation of Compliance with the APSA Principles and Guidance in
Human Subjects Research}\label{sec:app_irb}

This research involved an online survey sent to people who had used the Canadian
Broadcasting Corporation-sponsored Vote Compass tool: a non-partisan electoral
education initiative that invited citizens to answer twenty policy questions to
place themselves in a policy space relative to the major political parties
(which had also completed the survey). Over 1 million Canadians visited the Vote
Compass website surrounding the May 2011 federal election. We contacted all of
the 80,000 or so respondents who agreed to be contacted for future studies and about
10 percent agreed to take our survey. The survey consent process and instrument
   and plans for protecting the personal information of the survey respondents
   were reviewed by the Institutional Review Board of the university of the
   investigators: they were not appreciably different from those used in online
   surveys in general.


